Regularity properties in the classification program for separable amenable C*-algebras

Abstract

We report on recent progress in the program to classify separable amenable C<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"><mml:semantics><mml:msup><mml:mi /><mml:mo>∗<!-- ∗ --></mml:mo></mml:msup><mml:annotation encoding="application/x-tex">^*</mml:annotation></mml:semantics></mml:math></inline-formula>-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Z"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {Z}</mml:annotation></mml:semantics></mml:math></inline-formula>-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program’s successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.